cbviota

Description: Change bound variables in a description binder. Usage of this theorem is discouraged because it depends on ax-13 . Use the weaker cbviotaw when possible. (Contributed by Andrew Salmon, 1-Aug-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cbviota.1	\|- ( x = y -> ( ph <-> ps ) )
	cbviota.2	\|- F/ y ph
	cbviota.3	\|- F/ x ps
Assertion	cbviota	\|- ( iota x ph ) = ( iota y ps )

Proof

Step	Hyp	Ref	Expression
1		cbviota.1	\|- ( x = y -> ( ph <-> ps ) )
2		cbviota.2	\|- F/ y ph
3		cbviota.3	\|- F/ x ps
4		nfv	\|- F/ z ( ph <-> x = w )
5		nfs1v	\|- F/ x [ z / x ] ph
6		nfv	\|- F/ x z = w
7	5 6	nfbi	\|- F/ x ( [ z / x ] ph <-> z = w )
8		sbequ12	\|- ( x = z -> ( ph <-> [ z / x ] ph ) )
9		equequ1	\|- ( x = z -> ( x = w <-> z = w ) )
10	8 9	bibi12d	\|- ( x = z -> ( ( ph <-> x = w ) <-> ( [ z / x ] ph <-> z = w ) ) )
11	4 7 10	cbvalv1	\|- ( A. x ( ph <-> x = w ) <-> A. z ( [ z / x ] ph <-> z = w ) )
12	2	nfsb	\|- F/ y [ z / x ] ph
13		nfv	\|- F/ y z = w
14	12 13	nfbi	\|- F/ y ( [ z / x ] ph <-> z = w )
15		nfv	\|- F/ z ( ps <-> y = w )
16		sbequ	\|- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
17	3 1	sbie	\|- ( [ y / x ] ph <-> ps )
18	16 17	bitrdi	\|- ( z = y -> ( [ z / x ] ph <-> ps ) )
19		equequ1	\|- ( z = y -> ( z = w <-> y = w ) )
20	18 19	bibi12d	\|- ( z = y -> ( ( [ z / x ] ph <-> z = w ) <-> ( ps <-> y = w ) ) )
21	14 15 20	cbvalv1	\|- ( A. z ( [ z / x ] ph <-> z = w ) <-> A. y ( ps <-> y = w ) )
22	11 21	bitri	\|- ( A. x ( ph <-> x = w ) <-> A. y ( ps <-> y = w ) )
23	22	abbii	\|- { w \| A. x ( ph <-> x = w ) } = { w \| A. y ( ps <-> y = w ) }
24	23	unieqi	\|- U. { w \| A. x ( ph <-> x = w ) } = U. { w \| A. y ( ps <-> y = w ) }
25		dfiota2	\|- ( iota x ph ) = U. { w \| A. x ( ph <-> x = w ) }
26		dfiota2	\|- ( iota y ps ) = U. { w \| A. y ( ps <-> y = w ) }
27	24 25 26	3eqtr4i	\|- ( iota x ph ) = ( iota y ps )

Metamath Proof Explorer

Theorem cbviota

Proof